Statement of Purpose: The analysis of relationship between Personal Care Products' Price Index statistics and statistics of mascara, perfume, shampoo, various cosmetics markets(Watsons and Sephora), discounts, and dolar from Google Trends. The reason why these parameters are chosen as follows:
1 - Personal Care Products: Mascara, Perfume and Shampoo
These three keywords are chosen to analyze how much they have an affect over Personal Care Products' Price Index since they are the most known and used personal care product types that have different purposes. Due to the different purposes they have, I assumed they might have different effects on the price index, so I chose three of them.
2 - Markets Specialized For Cosmetics: Watsons and Sephora
These two keywords are chosen to analyze if the search amount of cosmetics stores have an affect over Personal Care Products' Price Index. My main question was " If Watsons has been searched over the Internet a lot, does this mean that Personal Care product sales started to increase or not? Does it have an effect over these products' price? " Since Watsons sells cheaper products compared to Sephora, I assumed this price range difference might have a different effect on price index. That's the reason why I chose both of these stores.
3 - Dolar
This keyword was chosen to primaly analyze the effect of dollar's value's rapid rise in our country on Personal Care products since most of these products are exported from other countries and thus, it is highly probable that the value of dollars affects the price of Personal Care Products.
4 - Discounts
This keyword is chosen to analyze if sales have an affect on overall price of personal care products.
Firstly, all necessary packages are installed. The data collected from EDVS and Google Trends had been put in a table format in Excel previously, now this Excel table is firstly converted into .csv format for R to read it. Then, the column names in this table are modified. Each set of data is plotted against time below.
hw1data <- read.csv("/Users/serratopaloglu/Desktop/360hw1/DENEME2.csv", sep=";")
colnames(hw1data) <- c("Date","Fiyat","Watsons","Maskara","Dolar","İndirim","Sampuan","Parfüm","Sephora")
head(hw1data)
| Date | Fiyat | Watsons | Maskara | Dolar | İndirim | Sampuan | Parfüm | Sephora |
|---|---|---|---|---|---|---|---|---|
| 2014-03 | 189.72 | 14 | 14 | 1 | 20 | 22 | 27 | 7 |
| 2014-04 | 191.74 | 10 | 15 | 1 | 20 | 23 | 29 | 6 |
| 2014-05 | 194.07 | 11 | 15 | 1 | 21 | 23 | 33 | 7 |
| 2014-06 | 196.17 | 14 | 14 | 1 | 21 | 19 | 29 | 7 |
| 2014-07 | 197.23 | 19 | 15 | 1 | 19 | 25 | 31 | 7 |
| 2014-08 | 198.97 | 17 | 14 | 1 | 19 | 26 | 30 | 7 |
plot(hw1data$Fiyat,type="l")
plot(hw1data$Watsons,type="l")
plot(hw1data$Maskara, type="l")
plot(hw1data$Dolar,type="l")
plot(hw1data$İndirim, type="l")
plot(hw1data$Sampuan,type="l")
plot(hw1data$Parfüm, type="l")
plot(hw1data$Sephora, type="l")
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Only from looking to these graphs, we can conclude that Fiyat(Personal Care Product Price Index) has an exponential trend over time which will need modeling later. All the independent variables also have a trend. Dolar, a positive small linear trend.Watsons, starting with a positive linear trend then to a decreasing linear trend.İndirim and Sampuan, starting linear then decreased to being constant. Maskara and Parfüm, linear. Sephora, a positive exponential trend.
Since all data obtained from Google Trends exist between 0 and 100, the dependent variable(Fiyat) is normalized. Now, we can check the summary of our data.
hw1data$Fiyat <- 100*((hw1data$Fiyat -min(hw1data$Fiyat))/(max(hw1data$Fiyat)-min(hw1data$Fiyat)))
summary(hw1data)
Date Fiyat Watsons Maskara
2014-03: 1 Min. : 0.000 Min. : 10.00 Min. : 14.00
2014-04: 1 1st Qu.: 2.933 1st Qu.: 25.00 1st Qu.: 34.00
2014-05: 1 Median : 11.912 Median : 47.00 Median : 62.00
2014-06: 1 Mean : 19.916 Mean : 43.82 Mean : 56.36
2014-07: 1 3rd Qu.: 21.720 3rd Qu.: 57.00 3rd Qu.: 77.00
2014-08: 1 Max. :100.000 Max. :100.00 Max. :100.00
(Other):115
Dolar İndirim Sampuan Parfüm
Min. : 1.00 Min. : 19.00 Min. : 19.00 Min. : 27.0
1st Qu.: 3.00 1st Qu.: 27.00 1st Qu.: 37.00 1st Qu.: 41.0
Median : 10.00 Median : 34.00 Median : 66.00 Median : 59.0
Mean : 11.69 Mean : 36.64 Mean : 56.94 Mean : 58.7
3rd Qu.: 15.00 3rd Qu.: 43.00 3rd Qu.: 73.00 3rd Qu.: 73.0
Max. :100.00 Max. :100.00 Max. :100.00 Max. :100.0
Sephora
Min. : 6.00
1st Qu.: 13.00
Median : 23.00
Mean : 23.88
3rd Qu.: 30.00
Max. :100.00
Now, it is necessary to examine the correlations between independent variable pairs to see if there is any one that could cause multicollienarity.
num.cols2 <- sapply(hw1data, is.numeric)
cor.data2 <- cor(hw1data[,num.cols2])
print(cor.data2)
Fiyat Watsons Maskara Dolar İndirim Sampuan Parfüm
Fiyat 1.0000000 0.4092595 0.7323660 0.3293462 0.2101800 0.6423663 0.8404616
Watsons 0.4092595 1.0000000 0.7809538 0.4211303 0.6676289 0.8391617 0.6922268
Maskara 0.7323660 0.7809538 1.0000000 0.4913568 0.5966897 0.9491590 0.9059376
Dolar 0.3293462 0.4211303 0.4913568 1.0000000 0.3869783 0.4899409 0.4485428
İndirim 0.2101800 0.6676289 0.5966897 0.3869783 1.0000000 0.6472773 0.5588179
Sampuan 0.6423663 0.8391617 0.9491590 0.4899409 0.6472773 1.0000000 0.8579284
Parfüm 0.8404616 0.6922268 0.9059376 0.4485428 0.5588179 0.8579284 1.0000000
Sephora 0.7652083 0.6553944 0.7623086 0.2946206 0.4144658 0.7761518 0.7835364
Sephora
Fiyat 0.7652083
Watsons 0.6553944
Maskara 0.7623086
Dolar 0.2946206
İndirim 0.4144658
Sampuan 0.7761518
Parfüm 0.7835364
Sephora 1.0000000
library(corrplot)
print(corrplot(cor.data2,method="color", tl.cex = 1.1, cl.cex=0.8))
print(corrplot(cor.data2,method="number", tl.cex = 1.1, cl.cex=0.8))
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$corr
Fiyat Watsons Maskara Dolar İndirim Sampuan Parfüm
Fiyat 1.0000000 0.4092595 0.7323660 0.3293462 0.2101800 0.6423663 0.8404616
Watsons 0.4092595 1.0000000 0.7809538 0.4211303 0.6676289 0.8391617 0.6922268
Maskara 0.7323660 0.7809538 1.0000000 0.4913568 0.5966897 0.9491590 0.9059376
Dolar 0.3293462 0.4211303 0.4913568 1.0000000 0.3869783 0.4899409 0.4485428
İndirim 0.2101800 0.6676289 0.5966897 0.3869783 1.0000000 0.6472773 0.5588179
Sampuan 0.6423663 0.8391617 0.9491590 0.4899409 0.6472773 1.0000000 0.8579284
Parfüm 0.8404616 0.6922268 0.9059376 0.4485428 0.5588179 0.8579284 1.0000000
Sephora 0.7652083 0.6553944 0.7623086 0.2946206 0.4144658 0.7761518 0.7835364
Sephora
Fiyat 0.7652083
Watsons 0.6553944
Maskara 0.7623086
Dolar 0.2946206
İndirim 0.4144658
Sampuan 0.7761518
Parfüm 0.7835364
Sephora 1.0000000
$corrPos
xName yName x y corr
1 Fiyat Fiyat 1 8 1.0000000
2 Fiyat Watsons 1 7 0.4092595
3 Fiyat Maskara 1 6 0.7323660
4 Fiyat Dolar 1 5 0.3293462
5 Fiyat İndirim 1 4 0.2101800
6 Fiyat Sampuan 1 3 0.6423663
7 Fiyat Parfüm 1 2 0.8404616
8 Fiyat Sephora 1 1 0.7652083
9 Watsons Fiyat 2 8 0.4092595
10 Watsons Watsons 2 7 1.0000000
11 Watsons Maskara 2 6 0.7809538
12 Watsons Dolar 2 5 0.4211303
13 Watsons İndirim 2 4 0.6676289
14 Watsons Sampuan 2 3 0.8391617
15 Watsons Parfüm 2 2 0.6922268
16 Watsons Sephora 2 1 0.6553944
17 Maskara Fiyat 3 8 0.7323660
18 Maskara Watsons 3 7 0.7809538
19 Maskara Maskara 3 6 1.0000000
20 Maskara Dolar 3 5 0.4913568
21 Maskara İndirim 3 4 0.5966897
22 Maskara Sampuan 3 3 0.9491590
23 Maskara Parfüm 3 2 0.9059376
24 Maskara Sephora 3 1 0.7623086
25 Dolar Fiyat 4 8 0.3293462
26 Dolar Watsons 4 7 0.4211303
27 Dolar Maskara 4 6 0.4913568
28 Dolar Dolar 4 5 1.0000000
29 Dolar İndirim 4 4 0.3869783
30 Dolar Sampuan 4 3 0.4899409
31 Dolar Parfüm 4 2 0.4485428
32 Dolar Sephora 4 1 0.2946206
33 İndirim Fiyat 5 8 0.2101800
34 İndirim Watsons 5 7 0.6676289
35 İndirim Maskara 5 6 0.5966897
36 İndirim Dolar 5 5 0.3869783
37 İndirim İndirim 5 4 1.0000000
38 İndirim Sampuan 5 3 0.6472773
39 İndirim Parfüm 5 2 0.5588179
40 İndirim Sephora 5 1 0.4144658
41 Sampuan Fiyat 6 8 0.6423663
42 Sampuan Watsons 6 7 0.8391617
43 Sampuan Maskara 6 6 0.9491590
44 Sampuan Dolar 6 5 0.4899409
45 Sampuan İndirim 6 4 0.6472773
46 Sampuan Sampuan 6 3 1.0000000
47 Sampuan Parfüm 6 2 0.8579284
48 Sampuan Sephora 6 1 0.7761518
49 Parfüm Fiyat 7 8 0.8404616
50 Parfüm Watsons 7 7 0.6922268
51 Parfüm Maskara 7 6 0.9059376
52 Parfüm Dolar 7 5 0.4485428
53 Parfüm İndirim 7 4 0.5588179
54 Parfüm Sampuan 7 3 0.8579284
55 Parfüm Parfüm 7 2 1.0000000
56 Parfüm Sephora 7 1 0.7835364
57 Sephora Fiyat 8 8 0.7652083
58 Sephora Watsons 8 7 0.6553944
59 Sephora Maskara 8 6 0.7623086
60 Sephora Dolar 8 5 0.2946206
61 Sephora İndirim 8 4 0.4144658
62 Sephora Sampuan 8 3 0.7761518
63 Sephora Parfüm 8 2 0.7835364
64 Sephora Sephora 8 1 1.0000000
$arg
$arg$type
[1] "full"
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$corr
Fiyat Watsons Maskara Dolar İndirim Sampuan Parfüm
Fiyat 1.0000000 0.4092595 0.7323660 0.3293462 0.2101800 0.6423663 0.8404616
Watsons 0.4092595 1.0000000 0.7809538 0.4211303 0.6676289 0.8391617 0.6922268
Maskara 0.7323660 0.7809538 1.0000000 0.4913568 0.5966897 0.9491590 0.9059376
Dolar 0.3293462 0.4211303 0.4913568 1.0000000 0.3869783 0.4899409 0.4485428
İndirim 0.2101800 0.6676289 0.5966897 0.3869783 1.0000000 0.6472773 0.5588179
Sampuan 0.6423663 0.8391617 0.9491590 0.4899409 0.6472773 1.0000000 0.8579284
Parfüm 0.8404616 0.6922268 0.9059376 0.4485428 0.5588179 0.8579284 1.0000000
Sephora 0.7652083 0.6553944 0.7623086 0.2946206 0.4144658 0.7761518 0.7835364
Sephora
Fiyat 0.7652083
Watsons 0.6553944
Maskara 0.7623086
Dolar 0.2946206
İndirim 0.4144658
Sampuan 0.7761518
Parfüm 0.7835364
Sephora 1.0000000
$corrPos
xName yName x y corr
1 Fiyat Fiyat 1 8 1.0000000
2 Fiyat Watsons 1 7 0.4092595
3 Fiyat Maskara 1 6 0.7323660
4 Fiyat Dolar 1 5 0.3293462
5 Fiyat İndirim 1 4 0.2101800
6 Fiyat Sampuan 1 3 0.6423663
7 Fiyat Parfüm 1 2 0.8404616
8 Fiyat Sephora 1 1 0.7652083
9 Watsons Fiyat 2 8 0.4092595
10 Watsons Watsons 2 7 1.0000000
11 Watsons Maskara 2 6 0.7809538
12 Watsons Dolar 2 5 0.4211303
13 Watsons İndirim 2 4 0.6676289
14 Watsons Sampuan 2 3 0.8391617
15 Watsons Parfüm 2 2 0.6922268
16 Watsons Sephora 2 1 0.6553944
17 Maskara Fiyat 3 8 0.7323660
18 Maskara Watsons 3 7 0.7809538
19 Maskara Maskara 3 6 1.0000000
20 Maskara Dolar 3 5 0.4913568
21 Maskara İndirim 3 4 0.5966897
22 Maskara Sampuan 3 3 0.9491590
23 Maskara Parfüm 3 2 0.9059376
24 Maskara Sephora 3 1 0.7623086
25 Dolar Fiyat 4 8 0.3293462
26 Dolar Watsons 4 7 0.4211303
27 Dolar Maskara 4 6 0.4913568
28 Dolar Dolar 4 5 1.0000000
29 Dolar İndirim 4 4 0.3869783
30 Dolar Sampuan 4 3 0.4899409
31 Dolar Parfüm 4 2 0.4485428
32 Dolar Sephora 4 1 0.2946206
33 İndirim Fiyat 5 8 0.2101800
34 İndirim Watsons 5 7 0.6676289
35 İndirim Maskara 5 6 0.5966897
36 İndirim Dolar 5 5 0.3869783
37 İndirim İndirim 5 4 1.0000000
38 İndirim Sampuan 5 3 0.6472773
39 İndirim Parfüm 5 2 0.5588179
40 İndirim Sephora 5 1 0.4144658
41 Sampuan Fiyat 6 8 0.6423663
42 Sampuan Watsons 6 7 0.8391617
43 Sampuan Maskara 6 6 0.9491590
44 Sampuan Dolar 6 5 0.4899409
45 Sampuan İndirim 6 4 0.6472773
46 Sampuan Sampuan 6 3 1.0000000
47 Sampuan Parfüm 6 2 0.8579284
48 Sampuan Sephora 6 1 0.7761518
49 Parfüm Fiyat 7 8 0.8404616
50 Parfüm Watsons 7 7 0.6922268
51 Parfüm Maskara 7 6 0.9059376
52 Parfüm Dolar 7 5 0.4485428
53 Parfüm İndirim 7 4 0.5588179
54 Parfüm Sampuan 7 3 0.8579284
55 Parfüm Parfüm 7 2 1.0000000
56 Parfüm Sephora 7 1 0.7835364
57 Sephora Fiyat 8 8 0.7652083
58 Sephora Watsons 8 7 0.6553944
59 Sephora Maskara 8 6 0.7623086
60 Sephora Dolar 8 5 0.2946206
61 Sephora İndirim 8 4 0.4144658
62 Sephora Sampuan 8 3 0.7761518
63 Sephora Parfüm 8 2 0.7835364
64 Sephora Sephora 8 1 1.0000000
$arg
$arg$type
[1] "full"
It can be clearly seen that a lot of pairs have correlation greater than 0.5 which is a sign of multicollienarity. The greatest correlation values can be detected in all the pairs with Maskara and Parfüm. Firstly, we will try to extract these two variables from our model, then check correlations again.
hwdata3 <- read.csv("/Users/serratopaloglu/Desktop/360hw1/deneme2.1.csv", sep=";")
colnames(hwdata3) <- c("Date","Fiyat","Watsons","Dolar","İndirim","Sampuan","Sephora")
hwdata3$Fiyat <- 100*((hwdata3$Fiyat -min(hwdata3$Fiyat))/(max(hwdata3$Fiyat)-min(hwdata3$Fiyat)))
head(hwdata3)
# Calculating Correlations on the new data set
num.cols2 <- sapply(hwdata3, is.numeric)
cor.data2 <- cor(hwdata3[,num.cols2])
print(cor.data2)
library(corrplot)
print(corrplot(cor.data2,method="color", tl.cex = 1.1, cl.cex=0.8))
print(corrplot(cor.data2,method="number", tl.cex = 1.1, cl.cex=0.8))
| Date | Fiyat | Watsons | Dolar | İndirim | Sampuan | Sephora |
|---|---|---|---|---|---|---|
| 2014-03 | 0.0000000 | 14 | 1 | 20 | 22 | 7 |
| 2014-04 | 0.1483255 | 10 | 1 | 20 | 23 | 6 |
| 2014-05 | 0.3194137 | 11 | 1 | 21 | 23 | 7 |
| 2014-06 | 0.4736135 | 14 | 1 | 21 | 19 | 7 |
| 2014-07 | 0.5514476 | 19 | 1 | 19 | 25 | 7 |
| 2014-08 | 0.6792131 | 17 | 1 | 19 | 26 | 7 |
Fiyat Watsons Dolar İndirim Sampuan Sephora Fiyat 1.0000000 0.4092595 0.3293462 0.2101800 0.6423663 0.7652083 Watsons 0.4092595 1.0000000 0.4211303 0.6676289 0.8391617 0.6553944 Dolar 0.3293462 0.4211303 1.0000000 0.3869783 0.4899409 0.2946206 İndirim 0.2101800 0.6676289 0.3869783 1.0000000 0.6472773 0.4144658 Sampuan 0.6423663 0.8391617 0.4899409 0.6472773 1.0000000 0.7761518 Sephora 0.7652083 0.6553944 0.2946206 0.4144658 0.7761518 1.0000000
Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in text.default(pos.xlabel[, 1], pos.xlabel[, 2], newcolnames, srt = tl.srt, : “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in text.default(pos.xlabel[, 1], pos.xlabel[, 2], newcolnames, srt = tl.srt, : “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in text.default(pos.xlabel[, 1], pos.xlabel[, 2], newcolnames, srt = tl.srt, : “font metrics unknown for Unicode character U+0130”Warning message in text.default(pos.ylabel[, 1], pos.ylabel[, 2], newrownames, col = tl.col, : “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in text.default(pos.ylabel[, 1], pos.ylabel[, 2], newrownames, col = tl.col, : “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”
$corr
Fiyat Watsons Dolar İndirim Sampuan Sephora
Fiyat 1.0000000 0.4092595 0.3293462 0.2101800 0.6423663 0.7652083
Watsons 0.4092595 1.0000000 0.4211303 0.6676289 0.8391617 0.6553944
Dolar 0.3293462 0.4211303 1.0000000 0.3869783 0.4899409 0.2946206
İndirim 0.2101800 0.6676289 0.3869783 1.0000000 0.6472773 0.4144658
Sampuan 0.6423663 0.8391617 0.4899409 0.6472773 1.0000000 0.7761518
Sephora 0.7652083 0.6553944 0.2946206 0.4144658 0.7761518 1.0000000
$corrPos
xName yName x y corr
1 Fiyat Fiyat 1 6 1.0000000
2 Fiyat Watsons 1 5 0.4092595
3 Fiyat Dolar 1 4 0.3293462
4 Fiyat İndirim 1 3 0.2101800
5 Fiyat Sampuan 1 2 0.6423663
6 Fiyat Sephora 1 1 0.7652083
7 Watsons Fiyat 2 6 0.4092595
8 Watsons Watsons 2 5 1.0000000
9 Watsons Dolar 2 4 0.4211303
10 Watsons İndirim 2 3 0.6676289
11 Watsons Sampuan 2 2 0.8391617
12 Watsons Sephora 2 1 0.6553944
13 Dolar Fiyat 3 6 0.3293462
14 Dolar Watsons 3 5 0.4211303
15 Dolar Dolar 3 4 1.0000000
16 Dolar İndirim 3 3 0.3869783
17 Dolar Sampuan 3 2 0.4899409
18 Dolar Sephora 3 1 0.2946206
19 İndirim Fiyat 4 6 0.2101800
20 İndirim Watsons 4 5 0.6676289
21 İndirim Dolar 4 4 0.3869783
22 İndirim İndirim 4 3 1.0000000
23 İndirim Sampuan 4 2 0.6472773
24 İndirim Sephora 4 1 0.4144658
25 Sampuan Fiyat 5 6 0.6423663
26 Sampuan Watsons 5 5 0.8391617
27 Sampuan Dolar 5 4 0.4899409
28 Sampuan İndirim 5 3 0.6472773
29 Sampuan Sampuan 5 2 1.0000000
30 Sampuan Sephora 5 1 0.7761518
31 Sephora Fiyat 6 6 0.7652083
32 Sephora Watsons 6 5 0.6553944
33 Sephora Dolar 6 4 0.2946206
34 Sephora İndirim 6 3 0.4144658
35 Sephora Sampuan 6 2 0.7761518
36 Sephora Sephora 6 1 1.0000000
$arg
$arg$type
[1] "full"
Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in text.default(pos.xlabel[, 1], pos.xlabel[, 2], newcolnames, srt = tl.srt, : “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in text.default(pos.xlabel[, 1], pos.xlabel[, 2], newcolnames, srt = tl.srt, : “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in text.default(pos.xlabel[, 1], pos.xlabel[, 2], newcolnames, srt = tl.srt, : “font metrics unknown for Unicode character U+0130”Warning message in text.default(pos.ylabel[, 1], pos.ylabel[, 2], newrownames, col = tl.col, : “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in text.default(pos.ylabel[, 1], pos.ylabel[, 2], newrownames, col = tl.col, : “conversion failure on 'İndirim' in 'mbcsToSbcs': dot substituted for <b0>”
$corr
Fiyat Watsons Dolar İndirim Sampuan Sephora
Fiyat 1.0000000 0.4092595 0.3293462 0.2101800 0.6423663 0.7652083
Watsons 0.4092595 1.0000000 0.4211303 0.6676289 0.8391617 0.6553944
Dolar 0.3293462 0.4211303 1.0000000 0.3869783 0.4899409 0.2946206
İndirim 0.2101800 0.6676289 0.3869783 1.0000000 0.6472773 0.4144658
Sampuan 0.6423663 0.8391617 0.4899409 0.6472773 1.0000000 0.7761518
Sephora 0.7652083 0.6553944 0.2946206 0.4144658 0.7761518 1.0000000
$corrPos
xName yName x y corr
1 Fiyat Fiyat 1 6 1.0000000
2 Fiyat Watsons 1 5 0.4092595
3 Fiyat Dolar 1 4 0.3293462
4 Fiyat İndirim 1 3 0.2101800
5 Fiyat Sampuan 1 2 0.6423663
6 Fiyat Sephora 1 1 0.7652083
7 Watsons Fiyat 2 6 0.4092595
8 Watsons Watsons 2 5 1.0000000
9 Watsons Dolar 2 4 0.4211303
10 Watsons İndirim 2 3 0.6676289
11 Watsons Sampuan 2 2 0.8391617
12 Watsons Sephora 2 1 0.6553944
13 Dolar Fiyat 3 6 0.3293462
14 Dolar Watsons 3 5 0.4211303
15 Dolar Dolar 3 4 1.0000000
16 Dolar İndirim 3 3 0.3869783
17 Dolar Sampuan 3 2 0.4899409
18 Dolar Sephora 3 1 0.2946206
19 İndirim Fiyat 4 6 0.2101800
20 İndirim Watsons 4 5 0.6676289
21 İndirim Dolar 4 4 0.3869783
22 İndirim İndirim 4 3 1.0000000
23 İndirim Sampuan 4 2 0.6472773
24 İndirim Sephora 4 1 0.4144658
25 Sampuan Fiyat 5 6 0.6423663
26 Sampuan Watsons 5 5 0.8391617
27 Sampuan Dolar 5 4 0.4899409
28 Sampuan İndirim 5 3 0.6472773
29 Sampuan Sampuan 5 2 1.0000000
30 Sampuan Sephora 5 1 0.7761518
31 Sephora Fiyat 6 6 0.7652083
32 Sephora Watsons 6 5 0.6553944
33 Sephora Dolar 6 4 0.2946206
34 Sephora İndirim 6 3 0.4144658
35 Sephora Sampuan 6 2 0.7761518
36 Sephora Sephora 6 1 1.0000000
$arg
$arg$type
[1] "full"
At this point, it is clear that Sampuan and Watsons variables continue to have high correlations with other independent variables. Since the assignment requires the student to establish the model of the dependent variable with at least 3 independent variables that have correlation less than 0.5 , I have decided to add new independent variables to my model. These variables were chosen as İşsizlik, Cilt Bakımı and Ramazan key words. My reasoning for adding these variables to my model can be ordered as follows:
1 - İşsizlik also known as Unemployement Rate, can affect the price index of Personal Care products. In a country with high unemployement rates and a worsening economy, the value of money decreases and people choose not to spend their money on personal care products to save money for their mandatory needs such as food, rent etc.
2 - Cilt Bakımı also known as Skin Care, is a relatively new term to our lives with various skin care products being manufactured and sold all around the world. With the high success of social media advertisements, people can go and purchase any skin care product that they see on online advertisements.
3- Ramazan is a holiday where major sales are conducted by cosmetic shops and drugstores such as Gratis, Watsons and Sephora similar to Black Friday. Thus, the days of this holiday has an affect on the sales of cosmetics and personal care products.
hw11data <- read.csv("/Users/serratopaloglu/Desktop/360hw1/deneme33.csv", sep=";")
colnames(hw11data) <- c("Date","Fiyat","Dolar","Sephora","Ramazan","İssizlik","CiltBakım")
head(hw11data)
| Date | Fiyat | Dolar | Sephora | Ramazan | İssizlik | CiltBakım |
|---|---|---|---|---|---|---|
| 2014-03 | 189.72 | 1 | 7 | 7 | 17 | 80 |
| 2014-04 | 191.74 | 1 | 6 | 9 | 18 | 77 |
| 2014-05 | 194.07 | 1 | 7 | 12 | 18 | 73 |
| 2014-06 | 196.17 | 1 | 7 | 31 | 19 | 73 |
| 2014-07 | 197.23 | 1 | 7 | 35 | 19 | 74 |
| 2014-08 | 198.97 | 1 | 7 | 6 | 17 | 72 |
plot(hw11data$Ramazan,type="l")
plot(hw11data$İssizlik,type="l")
plot(hw11data$CiltBakım, type="l")
Warning message in title(...): “conversion failure on 'hw11data$İssizlik' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(...): “conversion failure on 'hw11data$İssizlik' in 'mbcsToSbcs': dot substituted for <b0>”
Warning message in title(...): “conversion failure on 'hw11data$CiltBakım' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(...): “conversion failure on 'hw11data$CiltBakım' in 'mbcsToSbcs': dot substituted for <b1>”
From the graphs above, we can see that Ramazan has a clear seasonality, İssizlik has a positive linear trend and CiltBakım seems to be close to stationary with no apparent trend or seasonality.
hw11data$Fiyat <- 100*((hw11data$Fiyat -min(hw11data$Fiyat))/(max(hw11data$Fiyat)-min(hw11data$Fiyat)))
summary(hw11data)
head(hw11data)
Date Fiyat Dolar Sephora
2014-03: 1 Min. : 0.000 Min. : 1.00 Min. : 6.00
2014-04: 1 1st Qu.: 2.933 1st Qu.: 3.00 1st Qu.: 13.00
2014-05: 1 Median : 11.912 Median : 10.00 Median : 23.00
2014-06: 1 Mean : 19.916 Mean : 11.69 Mean : 23.88
2014-07: 1 3rd Qu.: 21.720 3rd Qu.: 15.00 3rd Qu.: 30.00
2014-08: 1 Max. :100.000 Max. :100.00 Max. :100.00
(Other):115
Ramazan İssizlik CiltBakım
Min. : 4.00 Min. : 17.0 Min. : 54.00
1st Qu.: 6.00 1st Qu.: 28.0 1st Qu.: 66.00
Median : 9.00 Median : 32.0 Median : 73.00
Mean : 18.38 Mean : 33.9 Mean : 72.96
3rd Qu.: 20.00 3rd Qu.: 39.0 3rd Qu.: 78.00
Max. :100.00 Max. :100.0 Max. :100.00
| Date | Fiyat | Dolar | Sephora | Ramazan | İssizlik | CiltBakım |
|---|---|---|---|---|---|---|
| 2014-03 | 0.0000000 | 1 | 7 | 7 | 17 | 80 |
| 2014-04 | 0.1483255 | 1 | 6 | 9 | 18 | 77 |
| 2014-05 | 0.3194137 | 1 | 7 | 12 | 18 | 73 |
| 2014-06 | 0.4736135 | 1 | 7 | 31 | 19 | 73 |
| 2014-07 | 0.5514476 | 1 | 7 | 35 | 19 | 74 |
| 2014-08 | 0.6792131 | 1 | 7 | 6 | 17 | 72 |
It is also useful to check our independent variables' scatterplots to detect any type of a pattern such as curvatures beforehand. There doesn't seem to be an exact curvature pattern in any pairs but Fiyat-Sephora pair. This pair will need a transformation to explain the curvature later on. Now, we will proceed to correlation calculation.
pairs(hw11data[, 2:6],
main = "Scatterplot Matrix")
Warning message in strwidth(labels, "user"): “conversion failure on 'İssizlik' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(labels, "user"): “conversion failure on 'İssizlik' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in text.default(x, y, txt, cex = cex, font = font): “conversion failure on 'İssizlik' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in text.default(x, y, txt, cex = cex, font = font): “conversion failure on 'İssizlik' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in text.default(x, y, txt, cex = cex, font = font): “font metrics unknown for Unicode character U+0130”
Now, with the newly chosen independent variables, their correlations will be checked. One can see a colored/numbered correlation matrix below.
num.cols2 <- sapply(hw11data, is.numeric)
cor.data2 <- cor(hw11data[,num.cols2])
print(cor.data2)
Fiyat Dolar Sephora Ramazan İssizlik CiltBakım Fiyat 1.0000000 0.329346173 0.7652083 0.225249744 0.1857805 0.17305213 Dolar 0.3293462 1.000000000 0.2946206 -0.008891708 0.2007231 0.01487221 Sephora 0.7652083 0.294620644 1.0000000 0.136772435 0.4201320 0.33193622 Ramazan 0.2252497 -0.008891708 0.1367724 1.000000000 0.1484542 0.06520440 İssizlik 0.1857805 0.200723060 0.4201320 0.148454204 1.0000000 0.27654518 CiltBakım 0.1730521 0.014872210 0.3319362 0.065204400 0.2765452 1.00000000
library(corrplot)
print(corrplot(cor.data2,method="color", tl.cex = 1.1, cl.cex=0.8))
print(corrplot(cor.data2,method="number", tl.cex = 1.1, cl.cex=0.8))
Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İssizlik' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'İssizlik' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'CiltBakım' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newrownames, cex = tl.cex): “conversion failure on 'CiltBakım' in 'mbcsToSbcs': dot substituted for <b1>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İssizlik' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'İssizlik' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion failure on 'CiltBakım' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in strwidth(newcolnames, cex = tl.cex): “conversion 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in 'mbcsToSbcs': dot substituted for <b1>”
$corr
Fiyat Dolar Sephora Ramazan İssizlik CiltBakım
Fiyat 1.0000000 0.329346173 0.7652083 0.225249744 0.1857805 0.17305213
Dolar 0.3293462 1.000000000 0.2946206 -0.008891708 0.2007231 0.01487221
Sephora 0.7652083 0.294620644 1.0000000 0.136772435 0.4201320 0.33193622
Ramazan 0.2252497 -0.008891708 0.1367724 1.000000000 0.1484542 0.06520440
İssizlik 0.1857805 0.200723060 0.4201320 0.148454204 1.0000000 0.27654518
CiltBakım 0.1730521 0.014872210 0.3319362 0.065204400 0.2765452 1.00000000
$corrPos
xName yName x y corr
1 Fiyat Fiyat 1 6 1.000000000
2 Fiyat Dolar 1 5 0.329346173
3 Fiyat Sephora 1 4 0.765208263
4 Fiyat Ramazan 1 3 0.225249744
5 Fiyat İssizlik 1 2 0.185780536
6 Fiyat CiltBakım 1 1 0.173052133
7 Dolar Fiyat 2 6 0.329346173
8 Dolar Dolar 2 5 1.000000000
9 Dolar Sephora 2 4 0.294620644
10 Dolar Ramazan 2 3 -0.008891708
11 Dolar İssizlik 2 2 0.200723060
12 Dolar CiltBakım 2 1 0.014872210
13 Sephora Fiyat 3 6 0.765208263
14 Sephora Dolar 3 5 0.294620644
15 Sephora Sephora 3 4 1.000000000
16 Sephora Ramazan 3 3 0.136772435
17 Sephora İssizlik 3 2 0.420132040
18 Sephora CiltBakım 3 1 0.331936218
19 Ramazan Fiyat 4 6 0.225249744
20 Ramazan Dolar 4 5 -0.008891708
21 Ramazan Sephora 4 4 0.136772435
22 Ramazan Ramazan 4 3 1.000000000
23 Ramazan İssizlik 4 2 0.148454204
24 Ramazan CiltBakım 4 1 0.065204400
25 İssizlik Fiyat 5 6 0.185780536
26 İssizlik Dolar 5 5 0.200723060
27 İssizlik Sephora 5 4 0.420132040
28 İssizlik Ramazan 5 3 0.148454204
29 İssizlik İssizlik 5 2 1.000000000
30 İssizlik CiltBakım 5 1 0.276545179
31 CiltBakım Fiyat 6 6 0.173052133
32 CiltBakım Dolar 6 5 0.014872210
33 CiltBakım Sephora 6 4 0.331936218
34 CiltBakım Ramazan 6 3 0.065204400
35 CiltBakım İssizlik 6 2 0.276545179
36 CiltBakım CiltBakım 6 1 1.000000000
$arg
$arg$type
[1] "full"
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in 'mbcsToSbcs': dot substituted for <b1>”
$corr
Fiyat Dolar Sephora Ramazan İssizlik CiltBakım
Fiyat 1.0000000 0.329346173 0.7652083 0.225249744 0.1857805 0.17305213
Dolar 0.3293462 1.000000000 0.2946206 -0.008891708 0.2007231 0.01487221
Sephora 0.7652083 0.294620644 1.0000000 0.136772435 0.4201320 0.33193622
Ramazan 0.2252497 -0.008891708 0.1367724 1.000000000 0.1484542 0.06520440
İssizlik 0.1857805 0.200723060 0.4201320 0.148454204 1.0000000 0.27654518
CiltBakım 0.1730521 0.014872210 0.3319362 0.065204400 0.2765452 1.00000000
$corrPos
xName yName x y corr
1 Fiyat Fiyat 1 6 1.000000000
2 Fiyat Dolar 1 5 0.329346173
3 Fiyat Sephora 1 4 0.765208263
4 Fiyat Ramazan 1 3 0.225249744
5 Fiyat İssizlik 1 2 0.185780536
6 Fiyat CiltBakım 1 1 0.173052133
7 Dolar Fiyat 2 6 0.329346173
8 Dolar Dolar 2 5 1.000000000
9 Dolar Sephora 2 4 0.294620644
10 Dolar Ramazan 2 3 -0.008891708
11 Dolar İssizlik 2 2 0.200723060
12 Dolar CiltBakım 2 1 0.014872210
13 Sephora Fiyat 3 6 0.765208263
14 Sephora Dolar 3 5 0.294620644
15 Sephora Sephora 3 4 1.000000000
16 Sephora Ramazan 3 3 0.136772435
17 Sephora İssizlik 3 2 0.420132040
18 Sephora CiltBakım 3 1 0.331936218
19 Ramazan Fiyat 4 6 0.225249744
20 Ramazan Dolar 4 5 -0.008891708
21 Ramazan Sephora 4 4 0.136772435
22 Ramazan Ramazan 4 3 1.000000000
23 Ramazan İssizlik 4 2 0.148454204
24 Ramazan CiltBakım 4 1 0.065204400
25 İssizlik Fiyat 5 6 0.185780536
26 İssizlik Dolar 5 5 0.200723060
27 İssizlik Sephora 5 4 0.420132040
28 İssizlik Ramazan 5 3 0.148454204
29 İssizlik İssizlik 5 2 1.000000000
30 İssizlik CiltBakım 5 1 0.276545179
31 CiltBakım Fiyat 6 6 0.173052133
32 CiltBakım Dolar 6 5 0.014872210
33 CiltBakım Sephora 6 4 0.331936218
34 CiltBakım Ramazan 6 3 0.065204400
35 CiltBakım İssizlik 6 2 0.276545179
36 CiltBakım CiltBakım 6 1 1.000000000
$arg
$arg$type
[1] "full"
It can be clearly seen that all independent variable pairs have correlations less than 0.5 which means that our current set for variables will be continued for the rest of the assignment. Let's fit our first model that does not contain any modifications on our independent and dependent variables.
model_2 <- lm(Fiyat ~ Dolar+CiltBakım+İssizlik+Sephora, data=hw11data)
summary(model_2)
residuals <- residuals(model_2)
plot(residuals)
acf(residuals)
hist(residuals, main = "Histogram of Residuals", xlab = "Residuals")
Call:
lm(formula = Fiyat ~ Dolar + CiltBakım + İssizlik + Sephora,
data = hw11data)
Residuals:
Min 1Q Median 3Q Max
-47.593 -7.990 -1.255 3.005 45.916
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.4095 11.3916 0.475 0.63577
Dolar 0.2383 0.1164 2.048 0.04281 *
CiltBakım -0.1425 0.1628 -0.875 0.38329
İssizlik -0.3625 0.1382 -2.623 0.00988 **
Sephora 1.4411 0.1179 12.223 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 15.1 on 116 degrees of freedom
Multiple R-squared: 0.6254, Adjusted R-squared: 0.6125
F-statistic: 48.41 on 4 and 116 DF, p-value: < 2.2e-16
First warning of our model's lack of validity is a relatively low value of R squared which is 0.6125 . Due to F statistics' low p value, we can conclude that our model is significant compared to a model with no slopes. And once t values are checked, we can see that Dolar, İssizlik and Sephora has a significant effect on Fiyat. Now, we will analyze the model's residuals. In the ACF plot it can be seen with the large correlation values that our model is not stationary yet and has some trend or seasonality that has not been explained. Let's see residual plots for each independent variable and try to detect which one is causing the curvature in residual plot.
library(lattice)
xyplot(partial_resid ~ Dolar, data = hw11data,
xlab = "Dolar", ylab = "Partial Residuals",
main = "Partial Residual Plot for Dolar")
xyplot(partial_resid ~ Sephora, data = hw11data,
xlab = "Sephora", ylab = "Partial Residuals",
main = "Partial Residual Plot for Sephora")
xyplot(partial_resid ~ Ramazan, data = hw11data,
xlab = "Ramazan", ylab = "Partial Residuals",
main = "Partial Residual Plot for Ramazan")
xyplot(partial_resid ~ CiltBakım, data = hw11data,
xlab = "CiltBakım", ylab = "Partial Residuals",
main = "Partial Residual Plot for CiltBakım")
xyplot(partial_resid ~ İssizlik, data = hw11data,
xlab = "İssizlik", ylab = "Partial Residuals",
main = "Partial Residual Plot for İssizlik")
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The residuals seem to be randomly scattered along the axis for CiltBakım and Ramazan. Which is a sign that they are not the variables causing the non linearity. The residuals seem to have a curvature for Dolar,Sephora and İssizlik variables. This curvature indicates that our model should be modified for an existing non linear relationship. Let's check ACF of Dolar,Sephora and İssizlik to see if there's any seasonality as well.
acf(hw11data$Dolar)
acf(hw11data$Sephora)
acf(hw11data$İssizlik)
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It can be clearly seen that the Sephora variable has the most repeating significant autocorrelation values. While Dolar and İssizlik do have significant spikes at the start, these spikes reduce as the lag gets larger. So, our first modification will be for Sephora variable.
# Modifying "Sephora" to make it stationary, we will use log transformation
hw11data_copy1$sepho <- log(hw11data_copy1$Sephora)
model_2_1 <- lm(Fiyat ~ Dolar+İssizlik+Ramazan+CiltBakım+sepho, data=hw11data_copy1)
summary(model_2_1)
Call:
lm(formula = Fiyat ~ Dolar + İssizlik + Ramazan + CiltBakım +
sepho, data = hw11data_copy1)
Residuals:
Min 1Q Median 3Q Max
-29.135 -11.610 -0.959 9.335 43.919
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -61.25478 12.12307 -5.053 1.66e-06 ***
Dolar 0.06356 0.12731 0.499 0.6185
İssizlik -0.65997 0.15426 -4.278 3.92e-05 ***
Ramazan 0.18217 0.07150 2.548 0.0122 *
CiltBakım -0.20222 0.17111 -1.182 0.2397
sepho 37.79591 3.46180 10.918 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 15.66 on 115 degrees of freedom
Multiple R-squared: 0.6005, Adjusted R-squared: 0.5832
F-statistic: 34.57 on 5 and 115 DF, p-value: < 2.2e-16
The R squared value did not increase that much, we will try to extract the mean from Sephora variable to ensure heteroscedasticity will be reduced to minimum.
hw11data_copy1$sepho <- log(hw11data_copy1$Sephora)
mean <- mean(hw11data_copy1$Sephora)
dif <- c(mean, hw11data_copy1$Sephora[1:120])
model_2_1 <- lm(Fiyat ~ Dolar+İssizlik+Ramazan+CiltBakım+sepho+dif, data=hw11data_copy1)
summary(model_2_1)
Call:
lm(formula = Fiyat ~ Dolar + İssizlik + Ramazan + CiltBakım +
sepho + dif, data = hw11data_copy1)
Residuals:
Min 1Q Median 3Q Max
-30.882 -10.756 -1.099 9.682 40.337
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -21.37188 13.22099 -1.617 0.10875
Dolar 0.10717 0.11471 0.934 0.35215
İssizlik -0.60325 0.13906 -4.338 3.12e-05 ***
Ramazan 0.17938 0.06426 2.791 0.00616 **
CiltBakım -0.34351 0.15606 -2.201 0.02974 *
sepho 20.19502 4.53892 4.449 2.02e-05 ***
dif 0.90006 0.16899 5.326 5.11e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 14.08 on 114 degrees of freedom
Multiple R-squared: 0.6801, Adjusted R-squared: 0.6633
F-statistic: 40.4 on 6 and 114 DF, p-value: < 2.2e-16
This is a better improvement compared to just taking Sephora's logarithm, but R squared did not improve greatly. Now, we will try to add the trend to our model.
hw11data_copy2 <- data.frame(hw11data)
hw11data_copy2$sepho <- log(hw11data_copy2$Sephora)
mean <- mean(hw11data_copy2$Sephora)
dif <- c(mean, hw11data_copy2$Sephora[1:120])
hw11data_copy2$trend <- 1:121
model_trend <- lm(Fiyat ~ sepho+dif+Dolar+Ramazan+İssizlik+CiltBakım+trend, data=hw11data_copy2)
summary(model_trend)
Call:
lm(formula = Fiyat ~ sepho + dif + Dolar + Ramazan + İssizlik +
CiltBakım + trend, data = hw11data_copy2)
Residuals:
Min 1Q Median 3Q Max
-23.376 -5.992 1.264 6.094 33.906
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 14.61105 11.25299 1.298 0.196789
sepho -6.12340 4.74802 -1.290 0.199797
dif 0.62587 0.13699 4.569 1.26e-05 ***
Dolar -0.26838 0.10074 -2.664 0.008846 **
Ramazan 0.06905 0.05227 1.321 0.189152
İssizlik -0.41139 0.11183 -3.679 0.000361 ***
CiltBakım -0.15716 0.12486 -1.259 0.210719
trend 0.59552 0.07073 8.419 1.34e-13 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 11.08 on 113 degrees of freedom
Multiple R-squared: 0.8034, Adjusted R-squared: 0.7912
F-statistic: 65.98 on 7 and 113 DF, p-value: < 2.2e-16
Significant improvement when trend is included in our model compared to the last try. Let's check our residuals.
residuals <- residuals(model_trend)
plot(residuals)
acf(residuals)
hist(residuals, main = "Histogram of Residuals", xlab = "Residuals")
Our residuals still have some curvature to them, to explain the non-linear relationship further we shall transform our dependent variable with taking its logarithm.
mean <- mean(hw11data_copy1$Sephora)
dif <- c(mean, hw11data_copy1$Sephora[1:120])
hw1_copy2$trend <- 1:121
model_new <- lm(log(Fiyat+1) ~ dif+log(Sephora)+Dolar+Ramazan+İssizlik+CiltBakım+trend, data=hw1_copy2)
summary(model_new)
residuals <- residuals(model_new)
plot(residuals)
acf(residuals)
hist(residuals, main = "Histogram of Residuals", xlab = "Residuals")
Call:
lm(formula = log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + Ramazan +
İssizlik + CiltBakım + trend, data = hw1_copy2)
Residuals:
Min 1Q Median 3Q Max
-0.38380 -0.07178 0.04057 0.08122 0.23338
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0922724 0.1261662 -0.731 0.4661
dif 0.0007937 0.0015359 0.517 0.6063
log(Sephora) 0.1251080 0.0532338 2.350 0.0205 *
Dolar -0.0016192 0.0011294 -1.434 0.1544
Ramazan 0.0003174 0.0005860 0.542 0.5892
İssizlik 0.0007209 0.0012539 0.575 0.5665
CiltBakım 0.0021150 0.0013999 1.511 0.1336
trend 0.0316119 0.0007930 39.862 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1243 on 113 degrees of freedom
Multiple R-squared: 0.9896, Adjusted R-squared: 0.989
F-statistic: 1539 on 7 and 113 DF, p-value: < 2.2e-16
Our R squared value has improved a lot, now it's 0.989 . Trend is the most significant coefficient due to its very high t-test value, log(Sephora) is the second most significant coefficient for this model. But the model's residuals seem to have a pattern which indicates our model is not valid to explain the dependent variable yet. Below there are tables regarding the pattern our residuals have with this model. It can be clearly seen that residuals do not form a perfect x=y line when put in the QQ plot.
plot(model_new)
Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + Ramazan + İssizlik + CiltB ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + Ramazan + İssizlik + CiltB ...' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + Ramazan + İssizlik + CiltB ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + Ramazan + İssizlik + CiltB ...' in 'mbcsToSbcs': dot substituted for <b0>”
Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + Ramazan + İssizlik + CiltB ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + Ramazan + İssizlik + CiltB ...' in 'mbcsToSbcs': dot substituted for <b0>”
Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + Ramazan + İssizlik + CiltB ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + Ramazan + İssizlik + CiltB ...' in 'mbcsToSbcs': dot substituted for <b0>”
res <- resid(model_new)
plot(fitted(model_new), res)
abline(0,0)
plot(model_new)
Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + dif_2 + sqrt(Sephora) + sqrt(Dolar) + Ramazan + İ ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + dif_2 + sqrt(Sephora) + sqrt(Dolar) + Ramazan + İ ...' in 'mbcsToSbcs': dot substituted for <b0>”
Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + dif_2 + sqrt(Sephora) + sqrt(Dolar) + Ramazan + İ ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + dif_2 + sqrt(Sephora) + sqrt(Dolar) + Ramazan + İ ...' in 'mbcsToSbcs': dot substituted for <b0>”
Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + dif_2 + sqrt(Sephora) + sqrt(Dolar) + Ramazan + İ ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + dif_2 + sqrt(Sephora) + sqrt(Dolar) + Ramazan + İ ...' in 'mbcsToSbcs': dot substituted for <b0>”
Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + dif_2 + sqrt(Sephora) + sqrt(Dolar) + Ramazan + İ ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + dif_2 + sqrt(Sephora) + sqrt(Dolar) + Ramazan + İ ...' in 'mbcsToSbcs': dot substituted for <b0>”
Lastly, we will model seasonality with dummy variables for Ramazan variable.
ramazan_month <- as.integer(hw1_copy2$Ramazan)
# Creating dummy variables for monthly seasonality of Ramazan
ramazan_dummies <- model.matrix(~ factor(ramazan_month) - 1)
mean <- mean(hw11data_copy1$Sephora)
dif <- c(mean, hw11data_copy1$Sephora[1:120])
hw1_copy2$trend <- 1:121
model_new2 <- lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + trend + ramazan_dummies, data = hw1_copy2)
summary(model_new2)
residuals <- residuals(model_new2)
plot(residuals)
acf(residuals)
hist(residuals, main = "Histogram of Residuals", xlab = "Residuals")
Call:
lm(formula = log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik +
CiltBakım + trend + ramazan_dummies, data = hw1_copy2)
Residuals:
Min 1Q Median 3Q Max
-0.32896 -0.03656 0.00000 0.06710 0.21555
Coefficients: (1 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0311183 0.2146015 -0.145 0.885
dif -0.0015498 0.0021092 -0.735 0.465
log(Sephora) 0.1193137 0.0740598 1.611 0.111
Dolar -0.0020203 0.0013243 -1.526 0.131
İssizlik 0.0016189 0.0017246 0.939 0.351
CiltBakım 0.0026893 0.0019433 1.384 0.171
trend 0.0321001 0.0010934 29.358 <2e-16
ramazan_dummiesfactor(ramazan_month)4 -0.2700212 0.1644941 -1.642 0.105
ramazan_dummiesfactor(ramazan_month)5 -0.0925087 0.1432923 -0.646 0.521
ramazan_dummiesfactor(ramazan_month)6 -0.0666610 0.1411373 -0.472 0.638
ramazan_dummiesfactor(ramazan_month)7 -0.1078359 0.1439485 -0.749 0.456
ramazan_dummiesfactor(ramazan_month)8 -0.1030499 0.1507079 -0.684 0.496
ramazan_dummiesfactor(ramazan_month)9 -0.0809728 0.1468484 -0.551 0.583
ramazan_dummiesfactor(ramazan_month)10 -0.0142236 0.1517534 -0.094 0.926
ramazan_dummiesfactor(ramazan_month)11 -0.0833827 0.1515152 -0.550 0.584
ramazan_dummiesfactor(ramazan_month)12 -0.1054143 0.1508380 -0.699 0.487
ramazan_dummiesfactor(ramazan_month)13 -0.0450791 0.1588644 -0.284 0.777
ramazan_dummiesfactor(ramazan_month)14 -0.0326692 0.1979646 -0.165 0.869
ramazan_dummiesfactor(ramazan_month)15 -0.1731772 0.1925048 -0.900 0.371
ramazan_dummiesfactor(ramazan_month)16 -0.1547670 0.1665472 -0.929 0.356
ramazan_dummiesfactor(ramazan_month)17 -0.0230339 0.1590060 -0.145 0.885
ramazan_dummiesfactor(ramazan_month)18 -0.1191052 0.1679914 -0.709 0.481
ramazan_dummiesfactor(ramazan_month)20 -0.0108306 0.1666677 -0.065 0.948
ramazan_dummiesfactor(ramazan_month)21 -0.0466558 0.1916894 -0.243 0.808
ramazan_dummiesfactor(ramazan_month)22 -0.0730686 0.1991194 -0.367 0.715
ramazan_dummiesfactor(ramazan_month)24 0.1050571 0.1954517 0.538 0.593
ramazan_dummiesfactor(ramazan_month)25 -0.0003082 0.1896278 -0.002 0.999
ramazan_dummiesfactor(ramazan_month)26 -0.0345431 0.1940476 -0.178 0.859
ramazan_dummiesfactor(ramazan_month)27 -0.1123293 0.1912221 -0.587 0.559
ramazan_dummiesfactor(ramazan_month)29 0.0371798 0.1951714 0.190 0.849
ramazan_dummiesfactor(ramazan_month)31 -0.1559437 0.1955560 -0.797 0.428
ramazan_dummiesfactor(ramazan_month)35 -0.1392621 0.1954568 -0.712 0.478
ramazan_dummiesfactor(ramazan_month)36 -0.1299921 0.1644468 -0.790 0.432
ramazan_dummiesfactor(ramazan_month)37 -0.0662022 0.1687721 -0.392 0.696
ramazan_dummiesfactor(ramazan_month)43 -0.3077132 0.1900516 -1.619 0.110
ramazan_dummiesfactor(ramazan_month)46 -0.0034943 0.1943879 -0.018 0.986
ramazan_dummiesfactor(ramazan_month)49 -0.0162728 0.1917226 -0.085 0.933
ramazan_dummiesfactor(ramazan_month)51 0.1089044 0.1946323 0.560 0.577
ramazan_dummiesfactor(ramazan_month)53 -0.0359948 0.1962905 -0.183 0.855
ramazan_dummiesfactor(ramazan_month)54 -0.0071624 0.1927425 -0.037 0.970
ramazan_dummiesfactor(ramazan_month)58 -0.0373902 0.1947130 -0.192 0.848
ramazan_dummiesfactor(ramazan_month)62 -0.2792626 0.1970132 -1.417 0.161
ramazan_dummiesfactor(ramazan_month)64 -0.1933806 0.1773463 -1.090 0.279
ramazan_dummiesfactor(ramazan_month)67 -0.0411942 0.1897969 -0.217 0.829
ramazan_dummiesfactor(ramazan_month)78 -0.1407847 0.1669260 -0.843 0.402
ramazan_dummiesfactor(ramazan_month)81 -0.0894808 0.1906576 -0.469 0.640
ramazan_dummiesfactor(ramazan_month)82 0.1469011 0.1975539 0.744 0.459
ramazan_dummiesfactor(ramazan_month)100 NA NA NA NA
(Intercept)
dif
log(Sephora)
Dolar
İssizlik
CiltBakım
trend ***
ramazan_dummiesfactor(ramazan_month)4
ramazan_dummiesfactor(ramazan_month)5
ramazan_dummiesfactor(ramazan_month)6
ramazan_dummiesfactor(ramazan_month)7
ramazan_dummiesfactor(ramazan_month)8
ramazan_dummiesfactor(ramazan_month)9
ramazan_dummiesfactor(ramazan_month)10
ramazan_dummiesfactor(ramazan_month)11
ramazan_dummiesfactor(ramazan_month)12
ramazan_dummiesfactor(ramazan_month)13
ramazan_dummiesfactor(ramazan_month)14
ramazan_dummiesfactor(ramazan_month)15
ramazan_dummiesfactor(ramazan_month)16
ramazan_dummiesfactor(ramazan_month)17
ramazan_dummiesfactor(ramazan_month)18
ramazan_dummiesfactor(ramazan_month)20
ramazan_dummiesfactor(ramazan_month)21
ramazan_dummiesfactor(ramazan_month)22
ramazan_dummiesfactor(ramazan_month)24
ramazan_dummiesfactor(ramazan_month)25
ramazan_dummiesfactor(ramazan_month)26
ramazan_dummiesfactor(ramazan_month)27
ramazan_dummiesfactor(ramazan_month)29
ramazan_dummiesfactor(ramazan_month)31
ramazan_dummiesfactor(ramazan_month)35
ramazan_dummiesfactor(ramazan_month)36
ramazan_dummiesfactor(ramazan_month)37
ramazan_dummiesfactor(ramazan_month)43
ramazan_dummiesfactor(ramazan_month)46
ramazan_dummiesfactor(ramazan_month)49
ramazan_dummiesfactor(ramazan_month)51
ramazan_dummiesfactor(ramazan_month)53
ramazan_dummiesfactor(ramazan_month)54
ramazan_dummiesfactor(ramazan_month)58
ramazan_dummiesfactor(ramazan_month)62
ramazan_dummiesfactor(ramazan_month)64
ramazan_dummiesfactor(ramazan_month)67
ramazan_dummiesfactor(ramazan_month)78
ramazan_dummiesfactor(ramazan_month)81
ramazan_dummiesfactor(ramazan_month)82
ramazan_dummiesfactor(ramazan_month)100
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Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1335 on 74 degrees of freedom
Multiple R-squared: 0.9922, Adjusted R-squared: 0.9873
F-statistic: 203.5 on 46 and 74 DF, p-value: < 2.2e-16
plot(model_new2)
Warning message: “not plotting observations with leverage one: 4, 5, 16, 17, 27, 28, 39, 40, 49, 50, 60, 64, 75, 87, 96, 97, 98, 107, 109, 110, 119, 120, 121”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <b1>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <b1>”
Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <b1>”Warning message: “not plotting observations with leverage one: 4, 5, 16, 17, 27, 28, 39, 40, 49, 50, 60, 64, 75, 87, 96, 97, 98, 107, 109, 110, 119, 120, 121”
Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <b0>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <c4>”Warning message in title(sub = sub.caption, ...): “conversion failure on 'lm(log(Fiyat + 1) ~ dif + log(Sephora) + Dolar + İssizlik + CiltBakım + tre ...' in 'mbcsToSbcs': dot substituted for <b1>”
After adding the dummy variables for seasonality in Ramazan, the residuals seem to be improved in terms of getting closer to having constant variance. Other than that, there doesn't seem to be any significant improvement to our model compared with the no seasonality added models. Trend coefficient is the most significant one, all the other coefficients are not significant for our model.
In conclusion, the problem for nonlinearity is not perfectly solved. Residuals' histogram shows that they are left skewed. Further modification is needed for our model to solve this skeweness. Also, the curvature in residual plot is still there even after various transformations and adding dummy variables. Even though, our model is shown to be significant compared to a model with no coefficients with its high F statistic value, and high R squared value, it is not a good model due to constant variance assumption for residuals being violated.